Moderating Factors of Immediate, Dynamic, and Long-run

Cross-Price Effects

Csilla Horváth and Dennis Fok

ERIM REPORT SERIES RESEARCH IN MANAGEMENT ERIM Report Series reference number ERS-2008-042-MKT Publication July 2008 Number of pages 39 Persistent paper URL http://hdl.handle.net/1765/12901 Email address corresponding author dfok@few.eur.nl Address Erasmus Research Institute of Management (ERIM)

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ABSTRACT AND KEYWORDS Abstract In this article the authors describe their comprehensive analysis of moderating factors of cross-

brand effects of price changes and contribute to the literature in five major ways. (1) They consider an extensive set of potential variables influencing cross-brand effects of price changes. (2) They examine moderators for the immediate as well as the dynamic cross-price effect. (3) They decompose price into regular and promotional price and study both cross-price effects separately. (4) They compare their findings with previous literature on the moderating factors of own-price effects to understand which factors influence own-price elasticity through affecting brand switching. (5) The authors use an advanced Bayesian estimation technique. The results show evidence of the neighborhood price effect and suggest that it is conditional on whether the promoted brand is priced above or below its competitor. The promoted brand's activities turn out to play a much more important role in determining the cross-price promotional effects than its competitor's similar activities. The authors outline conditions when cross-brand post-promotion dips tend to occur. Finally, they argue that the brand choice portion of the overall own-brand effect of a promotion depends on the brand's marketing strategy and on category-specific characteristics.

Free Keywords cross-price elasticity, asymmetry, dynamic effects, hierarchical Bayes

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Moderating factors of immediate, dynamic, and

long-run cross-price effects

Csilla Horvatha and Dennis Fokb

aInstitute for Management Research, Radboud University Nijmegen,

PO Box 9108, 6500 HK Nijmegen, The Netherlands

bErasmus Research Institute of Management,

and Econometric Institute, Erasmus University Rotterdam,

PO Box 1738, 3000 DR Rotterdam, The Netherlands

July 23, 2008

ABSTRACT

In this article the authors describe their comprehensive analysis of moderating factors

of cross-brand effects of price changes and contribute to the literature in five major ways.

(1) They consider an extensive set of potential variables influencing cross-brand effects of

price changes. (2) They examine moderators for the immediate as well as the dynamic

cross-price effect. (3) They decompose price into regular and promotional price and study

both cross-price effects separately. (4) They compare their findings with previous litera-

ture on the moderating factors of own-price effects to understand which factors influence

own-price elasticity through affecting brand switching. (5) The authors use an advanced

Bayesian estimation technique. The results show evidence of the neighborhood price ef-

fect and suggest that it is conditional on whether the promoted brand is priced above

or below its competitor. The promoted brand’s activities turn out to play a much more

important role in determining the cross-price promotional effects than its competitor’s

similar activities. The authors outline conditions when cross-brand post-promotion dips

tend to occur. Finally, they argue that the brand choice portion of the overall own-brand

effect of a promotion depends on the brand’s marketing strategy and on category-specific

characteristics.

Key words: Cross-price elasticity; Asymmetry; Dynamic effects; Hierarchical Bayes;

1

1 Introduction

For management purposes knowledge of cross-price effects is especially important. First

of all, managers need to know which rival brands have the strongest impact on their

sales. Second, they want to identify the competing brands that are affected the most

by their price changes and consequently from which competitor they should anticipate

reactions. As a next step, it is beneficial for them to recognize whether and how the cross-

price effects can be moderated by their and their competitors’ marketing activities and

positioning decisions. Finally, an interesting question is whether general characteristics

of the product category can explain the degree at which their sales are affected by other

brands’ price changes and their influence on others’ sales.

Patterns of cross-price effects have also been in the center of marketing research (e.g.,

Blattberg and Wisniewski, 1989; Bronnenberg and Wathieu, 1996; Sethuraman and Srini-

vasan, 2002). These patterns help understand brand price competition and market struc-

ture, thereby forces guiding pricing strategies (Sethuraman et al., 1999).

In this study we provide a comprehensive investigation of the moderating factors of

cross-price brand effects. More specifically, we focus on the moderating factors for the

immediate and the dynamic impact of price changes of a brand on its rivals’ sales. With

the dynamic impact we mean the impact of a current marketing action on future periods.

For a temporary price promotions this can be measured by the cumulative effect over the

future periods. For a permanent change in price, such as the adjustment of regular price

level, we measure the permanent impact, that is the impact on periods far in the future.

We contribute to the literature in five major ways: (i) we extend the list of studied

moderating variables for cross-price effects; (ii) we look at immediate and dynamic cross-

price effects; (iii) we decompose price into the regular price and the promotional price

and consider their cross effects separately; (iv) we compare the moderators of cross-price

elasticities with the moderators for the own-price effects found in Fok et al. (2006), and (v)

we estimate the cross-price effects and the impact of moderating variables simultaneously

in an efficient manner using a Hierarchical Bayes model. Below we further elaborate on

these contributions.

First, we consider several category-specific factors and separate their effects from the

influence of similar brand-specific variables. Furthermore, in addition to providing new

empirical evidence on the neighborhood and the asymmetric price effects (e.g., Blattberg

and Wisniewski, 1989; Bronnenberg and Wathieu, 1996; Sethuraman and Srinivasan, 2002)

2

we investigate whether the neighborhood effect depends on which of the two brands is

priced above. That is, we allow for asymmetry in the neighborhood effect. Regarding the

brand-specific covariates, an interesting question is which brand’s characteristics play a

more important role in shaping the cross-price effects; those of the “victim brand”, i.e.,

the brand whose sales are affected by the price changes of the other brand, or those of the

“attacker brand”, i.e., the brand that changes its price. Note that although these terms

clearly give the direction of the impact, it is not a priori known that the “victim brand”

will actually loose sales. The cross-brand effect could be zero or in special cases even

negative. Nonetheless, we will use the terms attacker and victim throughout this paper

to denote the two brands involved in a specific cross-brand effect.

Second, we examine the moderators for the immediate effects and the dynamic cross-

price effects. So far, the literature on cross-brand effects of price promotions focused on

the immediate or the long-run impact and did not consider the possibility of temporary

dynamic effects. The impact of price promotions of competing brands is probably not

limited to the immediate effect. For example, purchase acceleration may result in extra

inventory of the promoted brand. This additional inventory preempts future purchases

of competing brands (Ailawadi et al., 2007). This will especially hold for the influence

of a high-priced brand on a lower-priced brand. If an expensive high quality product is

for sale customers who, due to their tight budget constraint, usually buy a cheaper lower

quality brand may be inclined to stockpile the product and enjoy its benefits in the coming

periods, thereby postponing their usual purchase. So, asymmetry with respect to price

may be more important when considering the dynamic effect of a price promotion than

in case of the immediate effect. Additionally, deeper discounts provide higher financial

gains when switching to the promoted brand and stockpiling it. Such promotions may

result in a larger dynamic effect of price promotion. These examples illustrate that certain

variables’ influence may differ concerning the immediate effect and the dynamic effect.

We investigate these differences in this article.

Third, we decompose price into regular and promotional price and analyze the cross-

brand effects of both. The literature on own-brand effects has shown that changes in

regular price and price promotions have quite different effects on sales (Bijmolt et al.,

2005; Bucklin and Gupta, 1999; Fok et al., 2006). This is also likely to be the case for

cross-brand effects. For the promotional price and the regular price the immediate effect

is clearly of interest. Concerning the dynamic effect, for regular price we focus on the

3

(moderating factors of the) permanent or long-run effect. In this case, the permanent

effect is the effect of the price change on competitors’ sales far into the future. With

respect to the price promotions, we consider the (moderating factors of the) cumulative

effect. The cumulative effect equals the sum of the promotional effects on competitors’

sales over all periods.

Fourth, we compare our findings with the results of Fok et al. (2006) on the moder-

ating factors of own-price effects to understand which factors influence own-price elas-

ticity through affecting brand switching. These results reveal whether the category inci-

dence/brand choice/quantity division of the total own-effects of promotions may depend

on the involved brands’ marketing strategies or category-specific characteristics. That is,

they provide further insights into the decomposition of promotional response (Bell et al.,

1999).

Finally, in our investigation we rely on an efficient econometric technique, the Hier-

archical Bayes (HB) - Error Correction Model (ECM). This model allows us to directly

estimate the potentially differing immediate and dynamic effects of price changes of a

brand on its rival brands’ sales and to simultaneously relate these effects to characteris-

tics of brands and categories. Within the model we explicitly distinguish between (cross)

promotional price elasticities and (cross) regular price elasticities.

In sum, our analysis provides important insights for brand managers of the attacker

and the victim brands and for marketing scholars on the determinants that moderate the

cross-brand effect of price changes.

The remainder of this paper is as follows. We first present a systematic discussion of

the literature on the considered moderating effects and outline our conceptual research

framework. Next, we specify a sales response model to measure and explain the differences

in the immediate and dynamic cross-brand effects of promotional and regular prices. In

Section 4 we present the results of or empirical application based on a scanner database

on weekly sales volumes of the largest four brands in 25 product categories of fast moving

consumer goods. We end the paper with a discussion and conclusions in Section 5.

2 Moderating factors of the cross-price effects

The conceptual framework that guides our research is depicted in Figure 1. In this

figure we relate the cross-price effect to brand variables, category variables and variables

4

describing the relative positioning of the two involved brands. More precisely, we focus

on the effect of price promotions and regular price changes of brand A on the sales of

the rival brand V in the short- and in the long-run. In the figure letter A refers to the

attacker and V to the victim brand.

Below we discuss the literature concerning the variables we consider in our empirical

section. For each characteristic we summarize the literature and, if possible, provide a

hypothesis for the expected moderating effect. However, in many cases, especially for the

dynamic effects, we cannot formulate such hypotheses based on the literature. In these

cases we search for empirical evidence on whether and how the considered characteristics

are related to the (dynamic) effects of price. To facilitate a direct comparisons with

findings on own-price effects, we keep our set of moderating factors as similar as possible

to that in Fok et al. (2006).

2.1 Relative positioning of brands.

An interesting and well-researched aspect of cross-price effects concern the relative (price)

positioning of the attacker versus the victim brand. Examples of this are the asymmetric

and the neighborhood cross-price effect (Blattberg et al., 1995; Bronnenberg and Wathieu,

1996; Sethuraman, 1995; Sethuraman et al., 1999; Sethuraman and Srinivasan, 2002).

Asymmetric price effect. Higher priced brands have been found to be perceived

as being of better quality. This implies that price promotions by a higher-priced (better

quality) brand will be more attractive for the lower-budget segment and therefore affect

lower-priced (lower quality) brands more than the other way around (Allenby and Rossi,

1991; Blattberg and Wisniewski, 1989; Sethuraman et al., 1999, etc.). While promotions

of high-priced brands draw sales from their own price-tier competitors and from the tier

below, lower-quality brands’ promotions rarely take sales from tiers above (Blattberg and

Wisniewski, 1989; Sethuraman, 1995). Asymmetry can also arise from a difference in the

composition of the customer base of the two brands: a high quality (high-priced) brand

captures more loyal customers than a low quality (cheap) one. This leads to a higher

substitution effect for the better quality brand. Further explanations have been provided

in, for example, Blattberg and Neslin (1989) and Hardie et al. (1993). At the same time,

Bronnenberg and Wathieu (1996) point out that this asymmetry only holds if and only if

the quality advantage of the higher priced (higher quality) brand is sufficiently large, in

comparison with its price premium.

5

Neighborhood price effect. According to the neighborhood price effect hypothesis,

brands whose prices are closer to each other usually have larger cross-price effects than

brands that have more dissimilar prices. For example, Kamakura and Russell (1989)

show that consumers tend to switch only among brands within a certain price range.

Additional empirical evidence on the neighborhood effect has been found in Rao (1991),

Russel (1992), Sethuraman (1995), and Sethuraman et al. (1999).

An interesting question is whether the neighborhood or the asymmetric price effect is

stronger and whether one amplifies the other. Sethuraman et al. (1999) investigate this

and find that the neighborhood price effect is stronger than the asymmetric price effect,

both when relying on elasticities as on absolute price effects. We furthermore investigate

whether the neighborhood effect depends on which of the two brands is priced above.

Overall, we expect that if the attacker is priced above the victim brand and if the

two brands are positioned closer to each other, the cross-price effect will be higher. We

furthermore anticipate the neighborhood effect to be stronger when the attacker is more

expensive than the victim brand.

Asymmetric and neighborhood size effects. Analogous to the relative price

positioning, the relative positioning of the brands with respect to size may also influence

their cross-price effect. We measure two aspects of the size difference. We investigate

whether brands that are similar in size tend to have larger or smaller cross-price effects

and whether larger brands tend to be less sensitive to smaller brands’ promotions or the

other way around. We use similar measures as for the price effects and call the two effects

the asymmetric and neighborhood size effect.

Literature shows that larger brands usually have higher brand awareness and better

brand salience. Their promotions are therefore easier noticed and identified by customers

(e.g., Rao and Miller, 1975). Additionally, according to the well-established “double

jeopardy” phenomenon smaller brands tend to attract less “loyalty” among their buyers

than large brands do among theirs (Ehrenberg et al., 1990; Martin, 1973). Based on such

considerations Kamakura and Russell (1989), Sethuraman and Srinivasan (2002), and

Sethuraman (1995) suggest that larger brands have a larger influence in a market, and

that their promotions can easily hurt smaller brands. On the other hand, larger brands

are also less vulnerable to smaller brands’ discounts. This shows the possible existence

of asymmetry and the neighborhood size effects. In our empirical study we capture these

effects and also include interaction between the two effects to test for non-linearity in the

6

neighborhood size effect.

2.2 Category-and Brand-Specific Characteristics

To study the moderating effect of marketing activities we look at the intensity of marketing

activities at the category level as well as at such intensity for a brand relative to its

category. For example, in some categories promotions may be more frequent or deeper

than in others, but within a category there may also be relevant differences in promotional

policies.

Frequency of price promotions. According to the theory of price consciousness

high frequency of price promotions in a category leads to more price conscious customers

and this induces them to purchase on deal (Fok et al., 2006; Kopalle et al., 1999; Mela

et al., 1998; Yoo et al., 2000). More price conscious consumers may be more likely to

switch between brands. Papatla and Krishnamurthi (1996) similarly find that increased

purchases using coupons erode brand loyalty and increase price sensitivity. These results

suggest that categories with a high frequency of promotional activities are likely to show

substantial brand-switching and therefore a high immediate and dynamic cross-brand

price elasticity.

Within a category, other processes may also play a role. Brands with relatively frequent

price promotions are often considered of lower quality. Consequently, their price discounts

will be less attractive than similar brands with infrequent promotions and will stimulate

less brand switching and less stockpiling behavior. Frequent discounts also lower the

perceived quality of the victim brand whose share therefore is expected to respond more

its competitors’ price discounts (see Fok et al., 2006, and the discussion above on the

asymmetric price effect). For the price promotion frequency of the victim other effects

may also play a role. Frequently promoted brands draw a larger proportion of price-

sensitive customers (Zenor et al., 1998) who are likely to switch away when competitive

brands are on sale. In contrast, Jedidi et al. (1999) point out that frequent promotions of

brands make it unnecessary for the (loyal) customers to switch brands since they know

that a deal on their favored brand will occur in the near future. Finally, Jedidi et al.

(1999) mention that frequent discounts may make customers more likely to stockpile

their favorite brand because they fulfill a greater portion of their demand in promoted

periods.

Overall, the above findings point towards that the frequency of price promotional

7

activity of the attacker brand reduces the immediate cross-brand price elasticity and may

have an further moderating impact on the cumulative effect. For the victim we find mixed

indications in the literature and hypothese that, on the whole, their promotion frequency

increases the cross-price effect.

Average depth of promotions. Mehta et al. (2003) find that frequent price pro-

motions with deep discounts lead to large consideration sets. This is confirmed by Teng

(2008) who point out that deep discounts may stimulate otherwise loyal customers to

try other brands. Positive brand experience may make these customers to also consider

the alternative brand in future periods. This indicates a high brand-switching probabil-

ity in frequently promoted categories and in categories that are characterized by deep

discounts. Additionally, deep discounts have been found to increase deal sensitivity (An-

derson and Simester, 2004) and the variability of brand sales within a category (Raju,

1992). We therefore expect higher cross-brand elasticities in categories characterized by

deeper discounts.

At the brand level, consumers may believe that deep discounts indicate lower quality

(Jedidi et al., 1999; Mela et al., 1998) and unjustly high margins. On the other hand,

deep price promotions provide a high financial incentive for buying additional items of the

discounted brand and thereby stockpiling the product. This may result in an own-brand

post-promotion dip and may induce a prolonged dip in the sales of competing brands as

well. So, based on the literature we anticipate that general usage of deeper discounts

of the attacker brand will moderate the immediate price elasticity. However, the sign of

the effect is difficult to predict. Deep discounts by the attacker are likely to amplify the

cumulative elasticity. Deep discounts by the victim brands are expected to increase the

immediate effect.

Frequency of feature and display activities. Features and displays are often

used to increase or maintain brand awareness. High brand awareness may increase the

likelihood that the brand will be in the consideration set of customers (Keller, 1993) who

will therefore be more inclined to notice price promotions of these brands and eventually

buy them. Several studies have shown that feature and display activities can be used

to form consideration sets (e.g., Bronnenberg and Wathieu, 1996; Fader and McAlister,

1990; Mehta et al., 2003). Furthermore, retailers and manufacturers often support price

promotions with feature and display activities driven by the belief that such combinations

will generate larger incremental sales response to the promotion (Zhang, 2006). In sum,

8

price promotions of attacker and the victim brands and categories with more frequent

display or feature activities are expected to have higher cross-price promotional elasticity.

2.3 Category-Specific Characteristics

In this section we discuss how additional category-characteristics can influence the cross-

brand effect of price changes.

Perishability. The perishable nature of a product category has been found to be an

important feature for the own-price effects, especially with respect to the willingness to

store and stockpile the product (Narasimhan et al., 1996; Raju, 1992; Fok et al., 2006).

In line with these arguments for the own-price effects, we expect that in a category with

non-perishable products a part of the larger own-effect may arise from brand switchers

who have a high propensity to stockpile the product. The larger inventory of the promoted

brand will preempt the purchase of other brands hereby reducing their sales in the near

future (Ailawadi et al., 2007) causing a larger direct cross-brand effect as well as a larger

dynamic (post-promotion) effect.

Utilitarian nature. Shoppers with a relatively low income constitute a large pro-

portion of the consumer population of necessity goods (Bell et al., 1999; Wakefield and

Inman, 2003). This group is more price sensitive and deal-prone. Additionally, in low-

involvement categories brand awareness may be a sufficient condition for brand choice

(Keller, 1993). These findings point towards more brand-switching and therefore larger

cross-brand price effects in more utilitarian categories.

Average budget share. The budget share of a category captures two separate

dimensions: the general price level of the category and the average quantity purchased

in a period. The customer base of more expensive categories consists of higher income

households in general. They are less price sensitive and also less inclined to switch from

their favorite brand when a competitor is on sale. At the same time, promotions in

categories with high average purchase quantity induce customers to switch between brands

and stockpile more (Fok et al., 2006; Mace and Neslin, 2004). It is therefore difficult to

make a prediction for the impact of this variable.

Competitive intensity. In the literature on cross-price effects one of the important

findings regarding the competitive intensity is that cross-price effects are stronger when

there are fewer competing brands in the product category (Sethuraman et al., 1999). In

the presence of several similar brands, brand switching is likely to be the dominant source

9

of variability in brand sales in a category (Raju, 1992). This is supported by Narasimhan

et al. (1996) who point out that brand proliferation can be a source of weakened brand

loyalty. Mace and Neslin (2004) and Ailawadi et al. (2007) on the other hand show

that a large number of competitors may as well imply a lot of differentiation. Product

differentiation makes brands less exposed to competitors’ actions (Narasimhan et al.,

1996) and leads to less brand-switching.

3 Methodology

To describe the dynamic pattern in sales of brands in a number of product categories

we specify a set of flexible vector error correction (VEC) models. Within the model, the

differences in parameters across categories are explained in a second level model. This

hierarchical model is inspired on Fok et al. (2006). The important difference is that in

this article we explicitly focus on the cross-price effects. So, we define moderating factors

for cross-brand price effects in the second level of the model.

Also in this section we use the terminology “attacker” and “victim” brand, that is,

when we consider the effect of the price of brand j on the performance of brand i, we will

call brand j the “attacker” and brand i the “victim”.

3.1 Hierarchical Bayes Analysis

Let Sct denote the Ic-dimensional vector of sales for category c in week t. The Ic-

dimensional vectors Xckt contain the k-th marketing mix variables for the brands in

category c in week t. We specify the following vector error-correction model for cate-

gory c:

∆ log Sct = µc +K∑

k=1

Ack∆ log Xckt + Πc(log Sc,t−1 −K∑

k=1

Bck log Xck,t−1) + εct, (1)

with εct ∼ N(0, Σc) for c = 1, . . . , C and t = 1, . . . , Tc. The matrix Πc measures the

speed of adjustment of the sales to the long-run steady state. The matrices Ack and Bck

contain own and cross-brand effects. The own effects are on the diagonal, the cross effects

correspond to the off-diagonal elements. In general, let Aij,ck denote the element of Ack

on row i and column j, this corresponds to the effect of the k-th marketing instrument of

brand j on brand i.

10

It is relatively straightforward to show that Aij,ck can be interpreted as the immediate

effect of a change in Xj,ckt on brand i. More formally, it holds that

∂Sict

∂Xjct

Xjckt

Sict

=∂ log Sict

∂ log Xjckt

= Aij,ck. (2)

Note that because of the log-log specification in (1), we can interpret these effects as

elasticities. The parameter Bij,ck can be interpreted as either the cumulative effect of a

temporary change in Xjckt on Sic,t+τ , summed over τ = 0, 1, . . . or as the long-run effect

of a permanent shift in Xjckt on the sales levels of brand i in market c, see also Fok et al.

(2006). A temporary change is considered to be a one time change in Xjckt only at time

t, at time t + 1 the variable has returned to its previous level. For a permanent change,

Xjckt increases at time t and stays at that level. In terms of partial effects or elasticities,

the cumulative effect equals

∞∑τ=0

∂Sict

∂Xjckt

Xjckt

Sict

=∞∑

τ=0

log Sict

log Xjckt

= Bij,ck, (3)

and the long-run effect equals

limτ→∞

∂Sic,t+τ

∂Xjck

Xjck

Sic,t+τ

=∂ log Sic,t+τ

∂ log Xjck

= Bij,ck assuming Xjck = Xjckt = Xjck,t+1 = . . . .

(4)

Our main interest here is in the cross-effects, for notational efficiency we collect all

cross-effects for category c and marketing instrument k in Ic · (Ic−1)-dimensional vectors.

The cross-effect elements from Ack are collected in αck, the elements from Bck in βck.

More precisely,

αck = (A12,ck, A13,ck, . . . , A1I,ck, A21,ck, A23,ck, . . . , A2I,ck, · · · , AI1,ck, AI2,ck, . . . , AI,I−1,ck)′

βck = (B12,ck, B13,ck, . . . , B1I,ck, B21,ck, B23,ck, . . . , B2I,ck, · · · , BI1,ck, BI2,ck, . . . , BI,I−1,ck)′.

(5)

Now αck captures all the immediate effects of a change in the k-th marketing instrument

of a brand in category c on another brand’s sales in the same category. βck refers to all

respective long-run or dynamic effects. Finally, we introduce two functions i(l) and j(l).

These two functions map the index of an element of αck or βck to the index of the victim

and attacker brand, respectively. As an example, i(2) = 1 and j(2) = 3, that is, the

second elasticity in αck corresponds to the cross elasticity of brand 3 on brand 1, see also

(5).

11

The immediate and dynamic elasticities are expected to differ across brands and across

categories. Some of these differences can be attributed to observable characteristics of the

category, the attacker brand and/or the victim brand. Examples of such characteristics

are, depth and frequency of promotion or the perishability of the product, as we have

discussed in Section 2. The earlier mentioned neighborhood effects indicate that the dif-

ference between the attacker and victim brand on some scale may be important. Another

part of the differences across elasticities cannot be explained. In sum, we propose to

describe the immediate and dynamic elasticity parameters for the promotional price and

the regular price1 by

αl,ck = θ10k + θ1

1k′zi(l),c + θ1

2k′zj(l),c + θ1

3k′g(zi(l),c, zj(l),c) + θ1

4k′zc + η1

l,ck (6)

βl,ck = θ20k + θ2

1k′zi(l),c + θ2

2k′zj(l),c + θ2

3k′g(zi(l),c, zj(l),c) + θ2

4k′zc + η2

l,ck (7)

where zi,c is an vector containing explanatory variables for brand i in category c, like

frequency and depth of promotion. The vector zc contains a number of variables on the

category level, like category expensiveness. The neighborhood and asymmetry effect is

captured by the function g(), which for example may give the distance between the brands.

In the empirical analysis below we specify

g(zi(l),c, zj(l),c) =

|zj(l),c − zi(l),c|1zj(l),c>zi(l),c

|zj(l),c − zi(l),c| × 1zj(l),c>zi(l),c

, (8)

where 1A denotes an indicator function that equals one if the condition A is true and

zero otherwise. In words the function g() contains the absolute difference between the

two brands, an indicator whether the attacker is bigger or more expensive than the victim

brand, and the interaction between these two variables. Note that we do not include these

non-linear effects for all brand characteristics. For notational convenience we have not

made this explicit in (6)-(8). The vectors θ1nk and θ2

nk, n = 1, . . . , 4 describe the effects of

the brand characteristics on the immediate and the dynamic elasticities, respectively. For

the error terms η1l,ck and η2

l,ck we assume multivariate normal distributions. More specifi-

cally, we assume these error terms to be uncorrelated across brands, and categories. We

1In the empirical section we will also include other marketing instruments. For these instruments we

do not have a second layer specification as in (6)-(7). We do allow the elasticities for these variables to

differ across brands.

12

do however allow for correlation in the error terms across the K marketing-mix variables,

that is, we assume that

η1l,c = (η1

l,c1, . . . , η1l,cK)′ ∼ N(0, Σ1

η) and

η2ij,c = (η2

l,c1, . . . , η2l,cK)′ ∼ N(0, Σ2

η).(9)

For the own effects we have a similar specification as in (6) and (7) to link the own

effects to brand and category characteristics. Of course in this case only characteristics of

the brand itself and the category play a role. As the focus here is not on the own effects

we do not discuss this further. More details can be found in Appendix A.

To estimate the parameters of our model, (1) with (6)–(8), we consider a Bayesian

approach. In Appendix A we present the full details of the Markov Chain Monte Carlo

simulation method that we apply.

4 Empirical results

We apply our model to explain differences in immediate and dynamic cross-brand effects

of promotional price and regular price on sales across brands and product categories in an

extensive dataset. In Section 4.1 we describe the available data and the product categories

we consider in our analysis. Section 4.2 contains the estimation results.

4.1 Data and Variables

We use the Dominick’s Finer Foods data set for our empirical analysis. This data has

been used and described extensively in, for example, Fok et al. (2006) and Srinivasan

et al. (2004). So, we refer the reader for detailed description of the data to these papers.

We specify 25 error-correction models as in (1) for the 25 FMCG product categories

contained in the data set. In each model, the dependent variable Sct consists of the total

weekly sales of the four largest brands in product category c. As explanatory variables

in the first level of the model we consider the marketing-mix variables, display, feature,

regular price and promotional price indexes. These models are linked together through

the second level equations describing the price effects. The parameters in all model

parts are estimated simultaneously in a Bayesian setting. We consider several variables

as moderating factors of the cross-price elasticities (for their theoretical discussion see

13

Section 2 and for our conceptual model framework Figure 1). A summary and the formal

definition of these variables can be found in Appendix B.

The original database only contains the actual price. We use the decomposition of

the actual price into regular and promotional price as suggested in Fok et al. (2006).

Additionally, we account for seasonality and special holidays and run unit-root tests on

the seasonally adjusted series according to Fok et al. (2006). In line with previous research

(e.g., Fok et al., 2006; Horvath et al., 2005; Nijs et al., 2001; Pauwels et al., 2002), our

unit-root analysis shows that all sales series are (trend) stationary, after correcting for

possible seasonality and possible breaks in the regular price series.

Our approach differs in several important aspects from Fok et al. (2006). The main

difference is of course in the object that we study. Here we focus on the cross-brand effects

of price changes. Therefore, we also have a substantially higher number of elasticities

to model. In the considered 25 categories with 4 competing brands there are (16 −4) · 25 = 300 cross-promotional effects and only 4 · 25 = 100 own-brand effects per

instrument. Additionally, when analyzing differences in the cross-brand effects we consider

characteristics of both the attacker and the victim brand.

4.2 Estimation results

We use Gibbs sampling as presented in the Appendix A to obtain insight into the pa-

rameter values. The posterior results below are based on 200,000 draws of which the first

100,000 are used as burn in. To remove correlation in the chain we only consider every

10th draw for the computation of the posterior results. Unreported plots of the draws of

the model parameters of the second layer (6) and (7) show that the Markov Chain has

converged.

First, we summarize the posterior means of the cross effects of the (log) promotional

price index and the log regular price in graphs. Figure 2 presents the distribution of the

posterior means of the immediate cross effects of price promotions and of regular price

changes, the cumulative cross effect of price promotions, and the long-run cross effect of

a regular price change. These histograms show the distribution of the posterior means of

all brand-pairs in the 25 categories. Furthermore, we calculate the number of significant

positive and negative cross effects (see Table 1). Here we loosely use the term significant

to indicate the zero is not contained in the 95% highest posterior density region of the

corresponding parameter.

14

For the promotional price most of the (short- and cumulative) cross-price effects are

positive as expected (84% of the 300 cases) and about half of these positive values are

significantly larger than zero. The few negative effects are relatively small in size and

mostly not significant. Van Heerde et al. (2003) provide a possible explanation for the

negative values. A promotion of a brand may remind some customers about the cate-

gory, but these customers buy their strongly preferred brand that is different from the

promoted brand. When turning to the cumulative cross-promotional effects, the number

of significant effects drop. We do not find a single significant negative parameter and only

95 (32%) significant positive mean parameters.

Comparing the histograms for the promotional and regular price actions, we observe

much more dispersion of the cross-effects of regular price changes than of price promotions,

both in the short- and in the long-run. This is similar to the findings of Fok et al. (2006) for

own elasticities. For the regular price the graphs of Figure 2 show much more dispersion

with several positive and negative values. The majority of the posterior means, though,

is positive. Table 1 shows that for about 35% of the brand combinations we find a

negative impact. However, of all immediate cross-regular price effects only about 15% are

significantly different from zero. Interestingly, we find much more (almost 30%) significant

mean parameters for the long-run effect, despite the smaller dispersion. This suggests that

the reduction of regular price of a brand may have serious consequences on the sales of

at least a few of its competing brands. This may be due to a change in brand image

or realization of increased customer value, which are important considerations when the

customer develops its consideration set and later makes the final purchase decision. It is

also important to note that although not many of the effects are significant statistically,

they may very well be economically significant. A reason for not finding many significant

effects can be the lack of (variance in the) data. The regular prices do not change as

frequently as the promotional price index. Therefore, estimating the effects becomes a

more complicated task.

Only by looking at the effects over all brands can we make generalizing statements

on the patterns and the relevance of the cross regular price effects. So, in order to

gain insight into the pattern of the cross-brand dynamic effects we look at the post

cross-promotion dip, using results for each brand in (the first layer of) our model. If

consumers stockpile when a product is on promotion, this may preempt purchases of

other brands at a later point in time. A promotion of a brand will then lead to an

15

immediate disadvantageous impact on other brands, but also a harmful impact on other

brands in some periods to come. The cumulative cross-price effect will in this case be

larger than the immediate impact. To measure this, we calculate the posterior probability

that the immediate effect is smaller than the corresponding cumulative/long-term effect

for each cross-elasticity. If we find a large probability, this is a sign that there is a “post

cross-promotional dip”. This probability is calculated as the average, over all draws in

the MCMC sampler, of the percentage of cross effects for which the immediate effect is

smaller than the cumulative/long-term effect. For the promotional price this probability

equals 0.543, for the regular price it is 0.460. This result provides no clear empirical

evidence of the cross-brand post-promotion dip or any systematic dynamic effect. In other

words, most of the cross-price effect happens at the moment of the price change. However,

when we only compare the immediate and dynamic/long-run cross-brand effects when the

attacker brand is priced above the victim brand, this probability increases to 0.596 for

the promotional price (and 0.481 for regular price). This supports the premise that when

an expensive high quality product is for sale customers, who otherwise buy a cheaper and

lower quality alternative, may decide to buy and even to stockpile the discounted brand.

As a result they may postpone their usual purchase of lower quality brand whose sales

will experience a cross-brand post-promotional dip. We find even stronger indication of

cross-brand post promotion dip when focusing on effects of price changes of brands that

use deeper than average promotions and are priced higher than the victim. In this case

the dynamic cross-brand effect is larger than the immediate effect in about 66 percent of

the cases (result not presented in the tables). Deeper discount provides higher financial

gains (and higher increase in customer value) that could induce customers to stockpile

the promoted better quality brand.

4.2.1 Moderating factors of the cross-brand promotional effects

We now turn to the findings based on the second layer of our model and discuss which of

the considered moderating factors explain differences in the cross-effects for promotional

price. Table 2 shows the corresponding posterior results. All brand and category char-

acteristics have been standardized. Thus, the intercept can be interpreted as the mean

effect across all brand combinations. The intercept estimates are smaller for the imme-

diate than for the cumulative cross-brand effect, providing an additional evidence for the

cross-brand post-promotional dip.

16

We find quite some factors to have a significant influence on the cross promotional

price effect. Our results indicate partial evidence for the neighborhood price effect in

the short-run. Brands that are closer together in terms of average price, tend to have

larger cross-price effects if the attacker brand is cheaper than the victim brand. How-

ever, when the attacker is cheaper, asymmetry seems to dominate and the cross-brand

response. Cross-brand price elasticity appears to increase with the price (quality) differ-

ence between these brands.2 The cumulative effects are also positively related to these

two phenomena, however, the parameter of the cumulative neighborhood effect is not

significantly different from zero. This is probably due to the increased uncertainty (and

standard deviations) about the effect of brand- and category-specific variables on the

cumulative cross-promotional price effects. The increased uncertainty about the moder-

ating factors’ influence on the cumulative promotional price effect as compared to the

immediate influence seems to be a general phenomenon.

Our results provide evidence for the neighborhood size effect as well, suggesting that

sales of brands with similar size tend to react more to each others’ price promotional

activities. This finding holds irrespectively of whether the attacker or the victim brand is

larger; we find no evidence for an asymmetric size effect. The neighborhood size effect is

stronger in the long run; the parameters we find are about twice as large and significant.

Thus the neighborhood effect not only holds for the impact of a competitor’s price on the

current period, but also for the future periods.

Concerning the use of promotions, the attacker brand’s activities play an important

role in determining the cross-price promotional elasticities, while the victim brand’s ac-

tivities do not seem to influence the cross-brand promotional effects. The attacker’s

promotion frequency reduces the impact of its price cuts on competing brands’ sales. As

discussed before, an explanation can be that brands that are promoted frequently are of-

ten considered to have lower quality, than similar brands that are rarely promoted. At the

same time, while frequent usage of promotions lowers immediate cross-brand elasticity, it

does not affect the cumulative effect.

Brands with deeper promotions in general seem to have a stronger immediate and

2We test the significance of the neighborhood price effect when the attacker is more expensive

than the victim brand by checking whether zero is contained in the highest posterior density region

of the sum of parameters of |Attacker Price Index − Victim Price Index| and |Attacker Price Index −Victim Price Index| × I[Attacker Price Index > Victim Price Index], the posterior mean of this effect

size is −0.041 + 0.061 = 0.020.

17

cumulative impact on other brands. These results suggest that the additional financial

incentive exceeds the effect of lower perceived quality due to deeper discounts. Not only

does it induce more immediate brand switching but also stronger dynamic effects.

A frequent use of displays by the attacker is positively related to the size of the

cross promotional price effect. This result also holds for the cumulative effect. This,

as outlined in Section 2 may be due to the fact that overall displays tend to rise brand

awareness that increases the probability that the brand will be in the consideration set

of customers (Keller, 1993). Notably the effect size is almost the same for the immediate

as the cumulative effect, this implies that the display usage has a very limited impact on

the effect of a price promotion on future sales of other brands.

We find that several category-specific variables shape the cross promotional price effect,

however, some only in the short-run. In markets with a high market concentration, we

find smaller short-term cross promotional price effects. In the long-term it is the other

competitiveness measure, price differentiation, that appears to reduce brand switching.

Indeed, in markets without fierce competition between brands, we would expect less

switching among brands. In line with Mehta et al. (2003), Mela et al. (1998), and Zenor

et al. (1998), we find that markets with more frequent use of price promotions or relatively

deep price promotions tend to have stronger cross promotional price effects. For the

frequency we only find an effect in the short run, for the depth of price promotions we

find a short run and a long run effect. Additionally, average depth of price promotions

within the category seems to be more influential with respect to the cross-brand effect

than the attacker’s depth of price promotion. Another interesting finding is the impact

on the cumulative effect that is about twice as large as on the immediate effect. This may

suggest that deep promotions seriously erode brand-loyalty; people tend to stay with the

brand they bought on sale till a new one is promoted.

Finally, we find that cross promotional price effects are larger in more utilitarian

(lower involvement) categories. In such categories consumers tend to be less brand loyal

and more likely to switch brands.

4.2.2 Moderating factors of the cross-brand effects of changes in regular price

While we find quite some significant moderating variables for the cross-regular price effect

in the short-run, there are none for its long-run effect. An explanation may be that these

moderators mainly affect the speed of realizing the permanent price change and the pace

18

at which customers act upon it. Although there are quite some significant long-run effects,

29% see Table 1, it turns out to be very difficult to explain the differences in these long-

run effects. In some cases the entire category will profit from a regular price decrease,

leading to negative cross-price effects. In other cases, we get positive cross price effects

indicating that competitors loose sales if a brand reduces its regular price.

For the immediate effect, we find that a small brand is more affected by a change in

regular price of a large brand, than the other way around. In accordance with our findings

on the promotional effects, we find that mainly the attacker brand’s activities influence the

cross-brand effect of regular price changes. However, this time we find that the attacker’s

promotional frequency increases the cross-price effect. A possible explanation is that if a

frequently promoted brand increases its regular price, more people will be switching away

searching for alternative brands that are of better price-quality ratio. While the attacker’s

feature frequency did not affect price promotional elasticity, it increases the regular price

effect. On the other hand, feature frequency of the category turns out to reduce the effect

of a regular price change.

An interesting finding is that category characteristics that are mostly independent

from any marketing activities play the most important moderating role in the case of

regular price. In utilitarian and perishable categories changes in regular price induce

higher brand switching, but again, only in the short-run.

4.2.3 Relation between the first and the second layer of the model

In Table 3 we present the (co)variances of the error terms for the cross price effects (9).

These results show that while we are able to explain a relatively large part of the variation

in the cross promotional price effects, for the cross regular price effects this is not the case.

Especially for the long run effect the variance of the unexplained part is large. This of

course could also be noted from the fact that none of the explanatory variables turned

out to be significant for the long-run cross regular price effect. Note that the size of

the variance should be judged relative to the observed variation in the price effects, see

Figure 2. Furthermore, we find that there are no significant correlations between the

unobserved components for the cross-promotional and cross-regular price effects.

19

4.2.4 Comparison of findings for own- and cross-brand effects

We compare our findings with the results of Fok et al. (2006) on the moderating factors

of own-price effects to understand which factors influence own-price elasticity through

affecting brand switching.

Many of the brand specific variables that influence the own effects of a price change

found in Fok et al. (2006) also explain the cross effects. Average depth of promotion and

display frequency of the brand amplifies own-brand effects and the cross-brand effects in

the short, and in the long-run as well. So, some of the additional own-brand effect due

to deeper promotions or more frequent use of displays of the attacker brand arises from

higher brand-switching. Feature frequency neither influences own-brand nor cross-brand

effects. While for the own effect promotion frequency only influences the cumulative effect,

for the cross-elasticity it only moderates the immediate effect.

At the category level, we find, similar to the brand-specific counterpart, that the

use of deeper promotions in a category increases the own-effect and induces higher brand

switching as well. At the same time, more frequent use of displays, with strong influence at

the brand level, does not have any significant influence at the category level. With respect

to competition we find that less concentrated categories with low price differentiation are

characterized by relatively large own-price promotional effects possibly due to the high

brand-switching.

For utilitarian nature and perishability of categories the results match the least with

respect to the own- and cross-brand promotional effect. While a price promotion in

utilitarian categories results in the same amount of incremental sales as in non-utilitarian

ones, a higher portion of the incremental sales will be due to brand switching for utilitarian

products. Moreover, more perishable categories are characterized by low immediate own-

effects and average dynamic own effect, possibly due to lack of incentives to stockpile.

This phenomenon, however, does not affect the cross-brand elasticities.

Now we turn to the comparison with respect to the moderation for own- and cross-

brand effects of regular price changes. The most intriguing finding is that whereas in Fok

et al. (2006) no brand-specific variables had significant influence on the own-effects, we

find some brand-specific variables that moderate the cross-brand effects. The attacker’s

strategy about the rate of price promotion and feature usage amplifies the immediate

cross-brand price effect of its regular price changes. Among the category-specific variables

perishability and average depth of promotions influence immediate cross-effect, but do not

20

affect own-brand elasticity. At the same time, price dispersion only seems to influence the

effect of regular price changes on own sales. The utilitarian nature of the category amplifies

both the own- and the cross-brand effects in the short-run, however only influences the

own-effect in the long-run.

These results suggest that the category incidence/brand choice/quantity division of

the total own effects of promotions is not unconditional; it’s brand choice portion depends

on the brands’ strategy about its marketing instruments and mostly on category-specific

characteristics. A full study of such a conditional decomposition is beyond the scope of

the present paper. However, our results do provide further insights into the decomposition

of promotional response (Bell et al., 1999).

5 Conclusions

Our study provides new insights into the variability of cross-price effects and addresses

a couple of questions raised in previous literature on cross-price elasticities. Through a

hierarchical Bayes Error Correction model for 25 different product categories, we obtained

a number a generalizing findings.

First of all, we find support for the neighborhood cross price promotional effect. How-

ever, we show that this effect only holds if the attacker is in general cheaper (and therefore

probably of lower quality) than the victim brand. In this case if the attacker’s price is

closer to the victim’s price, it’s price promotion induces more customers of the victim

brand to switch. However, when the attacker is more expensive, we do not find evidence

for the neighborhood effect. So, the asymmetry plays a moderating role for the neighbor-

hood effect of price promotions. We also find evidence of neighborhood-size effects in the

short- and in the long-run as well.

Secondly, an interesting finding is that, among the brand-specific variables, activities

of the attacker brand significantly influence the cross-brand promotional price and cross-

brand regular price effects. The variables associated with the victim brand turn out not

to influence the cross-brand price effects. Most of the significant attacker characteristics

coincide with the own brand-specific variables that have been found to influence own-

brand elasticities significantly in Fok et al. (2006). Apparently, some of these earlier

established effects arise from higher brand-switching.

Furthermore, we find that an important factor influencing the cross-brand price effect

21

is whether the product category is utilitarian. Both the cross-brand promotional price

and the cross-brand effects of regular changes are significantly larger in more utilitar-

ian (lower involvement) categories. Additionally, the utilitarian nature seems to be the

most important moderator for the regular price, followed by whether the products in the

category are perishable or not.

Our study can be extended in several ways. First of all, we restricted our analysis to

the cross-price effects within a category. Analysis of possible moderating factors for cross-

price effects across categories, and hence considering the possibility of complementarity

in addition to substitutability, would bring new insight to the literature of cross-price

elasticities that would be useful for researchers and practitioners as well. However, one

should not expect to find many significant moderating effects in this case. The cross-effects

across categories will very likely be even smaller than within a category.

The investigation of cross-brand post-promotional effects seems to require further in-

vestigation as well. We showed that if the attacker brand is priced above the victim brand

and if it uses deep promotions, situation that indicates cross-brand post-promotion dip

is likely to occur. There may be situations when such dynamic cross-brand effects arise.

The investigation of these conditions would be interesting for marketing theory and prac-

tice. An interesting line of research would be to reveal the conditions under which such

cross-brand price promotion effects arise.

As we discussed in our section about the empirical analysis, the insignificant cross-

brand effects of the immediate and long-run effects of regular price changes, and therefore

the insignificant effect of the moderating factors on these elasticities, may be due to the

low volatility of the regular price variables in our dataset. Data with a larger longitudinal

dimension and with more regular price changes would facilitate the understanding of

cross-brand effects of changes in regular price.

Our analysis concerned FMCGs. An important question would be whether our findings

could be generalized over other categories, to durable products and/or to categories that

induce higher involvement of the customers.

22

Figure 1: Our conceptual research framework

Figure 2: Histogram of cross price effects

−0.50 −0.25 0.00 0.25 0.50 0.75

0.5

1.0

1.5

2.0Cumulative cross promotional price effect

−0.50 −0.25 0.00 0.25 0.50 0.75

1

2

Immediate cross promotional price effect

−2.5 0.0 2.5 5.0

0.2

0.4

Long run cross regular price effect

−2.5 0.0 2.5 5.0

0.1

0.2

Immediate cross regular price effect

23

Table 1: Counts related to the sign and significance of the cross-price effects per

brand combination. In total there are 25× 4× 3 = 300 cross-price effects

Promotional Regular

price price

Immediate effect

Significant and negative 10 2

Not significant and negative 39 102

Not significant and positive 119 157

Significant and positive 132 39

Cumulative/Long run effect

Significant and negative 0 24

Not significant and negative 52 85

Not significant and positive 153 128

Significant and positive 95 63

Cumulative/Long run effect − Immediate effect

Significant and negative 3 33

Not significant and negative 134 127

Not significant and positive 155 124

Significant and positive 8 16

Note: “Significant” should be read as: zero is not contained in the 95% HPD

region.

24

Tab

le2:

Pos

teri

orm

eans

(and

stan

dar

ddev

iati

ons)

for

cros

spri

ceeff

ect

moder

ator

s

Var

iabl

e1C

ross

Pro

mot

iona

lP

rice

Effe

ctC

ross

Reg

ular

Pri

ceE

ffect

Imm

edia

teeff

ect

Cum

ulat

ive

effec

tIm

med

iate

effec

tLon

gR

un

Inte

rcep

t0.

166

(0.0

43)*

**0.

252

(0.0

61)*

**0.

142

(0.4

64)

0.10

3(0

.235

)

|Att

acke

rP

rice

Inde

x−

Vic

tim

Pri

ceIn

dex|

-0.0

41(0

.021

)**

-0.0

48(0

.030

)0.

106

(0.2

47)

-0.0

05(0

.115

)

|Att

acke

rB

rand

Size−

Vic

tim

Bra

ndSi

ze|

-0.0

29(0

.018

)*-0

.060

(0.0

26)*

*0.

231

(0.1

71)

0.05

7(0

.106

)

I[A

ttac

ker

Pri

ceIn

dex

>V

icti

mP

rice

Inde

x]0.

035

(0.0

42)

0.06

1(0

.060

)-0

.228

(0.4

42)

0.02

9(0

.236

)

I[A

ttac

ker

Bra

ndSi

ze>

Vic

tim

Bra

ndSi

ze]

0.06

1(0

.041

)-0

.046

(0.0

57)

0.84

8(0

.439

)*-0

.116

(0.2

31)

|Att

acke

rP

rice

Inde

x−

Vic

tim

Pri

ceIn

dex|

0.06

1(0

.027

)**

0.08

8(0

.041

)**

-0.0

34(0

.362

)0.

061

(0.1

57)

×I[A

ttac

ker

Pri

ceIn

dex

>V

icti

mP

rice

Inde

x]

|Att

acke

rB

rand

Size−

Vic

tim

Bra

ndSi

ze|

0.02

9(0

.025

)0.

043

(0.0

38)

-0.3

98(0

.310

)0.

230

(0.1

45)

×I[A

ttac

ker

Bra

ndSi

ze>

Vic

tim

Bra

ndSi

ze]

Vic

tim

rela

tive

pric

epr

omot

ion

freq

uenc

y0.

023

(0.0

18)

0.02

4(0

.028

)0.

154

(0.2

10)

0.01

9(0

.106

)

Vic

tim

rela

tive

dept

hof

pric

epr

omot

ions

0.00

4(0

.012

)-0

.005

(0.0

17)

0.07

1(0

.222

)-0

.046

(0.0

73)

Vic

tim

rela

tive

feat

ure

freq

uenc

y-0

.010

(0.0

15)

0.01

3(0

.025

)0.

002

(0.1

70)

0.02

6(0

.097

)

Vic

tim

rela

tive

disp

lay

freq

uenc

y0.

019

(0.0

13)

0.01

4(0

.019

)0.

016

(0.1

59)

-0.0

48(0

.084

)

Att

acke

rre

lati

vepr

ice

prom

otio

nfr

eque

ncy

-0.0

42(0

.019

)**

0.01

1(0

.029

)0.

607

(0.3

31)*

-0.0

28(0

.111

)

Att

acke

rre

lati

vede

pth

ofpr

ice

prom

otio

ns0.

021

(0.0

12)*

0.02

6(0

.015

)*0.

327

(0.2

77)

0.07

5(0

.077

)

Att

acke

rre

lati

vefe

atur

efr

eque

ncy

-0.0

09(0

.017

)-0

.026

(0.0

23)

0.54

0(0

.308

)*-0

.060

(0.0

96)

Att

acke

rre

lati

vedi

spla

yfr

eque

ncy

0.05

5(0

.017

)***

0.05

6(0

.025

)**

-0.2

43(0

.249

)-0

.039

(0.0

92)

Cat

egor

ypr

ice

disp

ersi

on-0

.018

(0.0

20)

-0.0

60(0

.029

)**

-0.1

13(0

.372

)-0

.077

(0.0

96)

Mar

ket

conc

entr

atio

nin

dex

-0.0

61(0

.018

)***

-0.0

16(0

.027

)-0

.539

(0.3

34)

-0.0

73(0

.099

)

Cat

egor

yex

pens

iven

ess

0.02

0(0

.020

)0.

044

(0.0

28)

0.12

1(0

.525

)0.

013

(0.1

13)

Cat

egor

ypr

ice

prom

otio

nfr

eque

ncy

0.05

8(0

.023

)**

0.01

1(0

.033

)-0

.303

(0.4

18)

0.14

4(0

.114

)

Cat

egor

yde

pth

ofpr

ice

prom

otio

ns0.

049

(0.0

17)*

**0.

071

(0.0

22)*

**-0

.029

(0.3

61)

0.04

3(0

.085

)

Cat

egor

yfe

atur

efr

eque

ncy

0.02

2(0

.023

)0.

047

(0.0

30)

-0.6

53(0

.367

)*-0

.058

(0.1

08)

Cat

egor

ydi

spla

yfr

eque

ncy

0.00

0(0

.019

)-0

.035

(0.0

29)

-0.1

29(0

.444

)-0

.061

(0.1

11)

Uti

litar

ian

0.08

0(0

.020

)***

0.07

2(0

.028

)**

1.17

9(0

.415

)***

-0.0

43(0

.098

)

Per

isha

bilit

y0.

014

(0.0

19)

0.02

6(0

.026

)0.

708

(0.3

61)*

-0.0

96(0

.096

)

***,

**,*

:99

%,9

5%,9

0%hi

ghes

tpo

ster

ior

dens

ity

regi

ons

dono

tco

ntai

nze

ro,re

spec

tive

ly.

1:

I[A

ttac

ker

Pri

ceIn

dex

>V

icti

mP

rice

Inde

x]is

anin

dica

tor

that

equa

lson

eif

the

atta

cker

bran

dis

mor

eex

pens

ive

than

the

vict

imbr

and.

25

Table 3: Error variances cross price effects (Σ1η and Σ2

η), posterior means with pos-

terior standard errors in parentheses

Immediate cross Immediate cross

promotional price regular price

Immediate cross promotional price effect 0.0272 -0.0006

(0.0036) (0.0131)

Immediate cross regular price effect -0.0006 0.2753

(0.0131) (0.2100)

Cumulative cross Long run cross

promotional price regular price

Cumulative cross promotional price 0.0301 0.0139

(0.0051) (0.0237)

Long run cross regular price 0.0139 1.1954

(0.0237) (0.1567)

26

A Bayesian estimation

In this appendix we discuss the details of the Bayesian analysis of our model. We consider some

technical details and we present all used conditional distributions. For the model analysis we

explicitly model the first observation, that is for the analysis we consider the exact likelihood

function. As for the first observation lags are not available, we put this observation equal to the

long-run equilibrium level, that is,

log Sc1 = −Π−1c µc +

K∑

k=1

Bck log Xck1 + εc1 (10)

with ε1c ∼ N(0, Vc), where Vc is the long-run variance. This variance can be obtained from

solving the system by repeated substitution and is given by Vc =∑∞

j=0 ΓjcΣc(Γ′c)j , where Γc =

I + Πc. The variance is finite if the eigenvalues of Γc are within the unit circle, that is, in case

of stationarity.

To derive the likelihood function, we summarize the elements of Ak and Bk which we relate

to the own effects, in the K-dimensional row vectors αic = [Aii,ck]Kk=1 and βic = [Bii,ck]Kk=1. The

equations we impose for the own effects can be written as

αic = Λ′1hic + ν1ic (11)

βic = Λ′2hic + ν2ic (12)

for i = 1, . . . , Ic, where hic denotes an L × 1 vector of explanatory variables. We specify ν1ic ∼

N(0,Σ1ν) and ν2

ic ∼ N(0, Σ2ν). We collect the cross effects over different marketing instruments

in the vectors αlc = (αlc1, . . . , αlcK)′ and βlc = (βlc1, . . . , βlcK)′. We compactly write the second

stage equations (6) and (7) as

αlc = Θ′1Zlc + η1

lc (13)

βlc = Θ′2Zlc + η2

lc (14)

where Zlc collects all variables in (6) and (7), and ηnlc = (ηn

lc1, . . . , ηnlcK)′, n = 1, 2.

The likelihood function of the model is given by

C∏

c=1

∫

αc,βc

∫

αc,βc

φ(εc1; 0, Vc)Tc∏

t=2

φ(εct; 0, Σc)Ic∏

i=1

φ(αic; Λ′1hic,Σ1ν)φ(βic; Λ′2hic, Σ2

ν)×

Ic(Ic−1)∏

l=1

φ(αlc; Θ′1Zic, Σ1

η)φ(βlc; Θ′2Zlc,Σ2

η)dαcdβcdαcdβc, (15)

where φ(x; µ,Σ) is the density function of the multivariate normal distribution with mean µ

and variance Σ evaluated at x, and where αc = (α′1c, . . . , α′Icc)

′, βc = (β′1c, . . . , β′Icc)

′, αc =

(α′1c, . . . , α′Ic(Ic−1),c)

′, and βc = (β′1c, . . . , β′Ic(Ic−1),c)

′.

27

To obtain posterior results, we use the Gibbs sampling technique of Geman and Geman

(1984) with data augmentation, see Tanner and Wong (1987). An introduction into the Gibbs

sampler can be found in Casella and George (1992), see also Smith and Roberts (1993) and

Tierney (1994). Latent variables are sampled alongside the model parameters. The Bayesian

analysis is largely based on uninformative priors for the model parameters. To improve conver-

gence of the MCMC sampler we impose inverted Wishart priors on the Σnη and Σn

ν , n = 1, 2

parameters with scale parameter κ1IK and degrees of freedom κ2. We set the value of κ1 to

1 and κ2 equal to K + 3 such that the influence of the prior on the posterior distribution is

marginal, see Hobert and Casella (1996) for a discussion.

Below we derive the full conditional posterior distributions of the model parameters and all

latent variables. In deriving the sampling distributions we build on the general results in Zellner

(1971, Chapter VIII), and those in Fok et al. (2006). Altough we use a slightly different notation

here, the sampling distributions for some parameters are actually exactly the same as in Fok et

al. (2006). For completeness, we however present all results here.

Sampling of ΠcΠcΠc

To sample Πc we follow the same approach as in Fok et al. (2006), that is we rely on a Metropolis-

Hastings [MH] sampler (Metropolis et al., 1953; Hastings, 1970). As the candidate we consider

the distribution that would result if we ignore the first observation, that is, we write (1) as

∆ log Sct − µc −K∑

k=1

Ack∆log Xckt = Πc(log Sc,t−1 −K∑

k=1

Bck∆log Xck,t−1) + εct. (16)

We summarize this equation as Yct = ΠcWct + εct, where Yct and Wct denote the corresponding

terms in (16). Ignoring the observation for t = 1 we would obtain a normal full conditional

posterior distribution with mean

Π′c =

(Tc∑

t=2

WctW′ct

)−1 (Tc∑

t=2

WctY′ct

)(17)

and variance

ΣΠ′c =

Σc ⊗

(Tc∑

t=2

WctW′ct

)−1 . (18)

This distribution is expected to be similar to the true posterior distribution, therefore it will

work well as a candidate for the M-H sampler. The sampled candidate is denoted by Πcandc . The

difference in the candidate and the target density is only in the first observation. This allows us

to write the acceptance-rejection probability as

φ(ε1c; 0, Vc)|Πc=Πcandc

φ(Πcand′c ; Π′c, ΣΠ′c)φ(Πold′

c ; Π′c, ΣΠ′c)

φ(ε1c; 0, Vc)|Πc=Πoldc

φ(Πold′c ; Π′c, ΣΠ′c)φ(Πcand′

c ; Π′c, ΣΠ′c)=

φ(ε1c; 0, Vc)|Πc=Πcandc

φ(ε1c; 0, Vc)|Πc=Πoldc

, (19)

28

where Πoldc denotes the previous draw and εc1 = log Sc1 + Π−1

c µc −∑K

k=1 Bck log Xck1, see Chib

and Greenberg (1995) for a similar approach in an exact likelihood analysis of an autoregressive

model.

Sampling of ΣcΣcΣc

Our treatment of the first observations also leads to a non standard sampling distribution for Σc.

We use the same procedure as for Πc, that is, we use a Metropolis-Hastings sampler where we

construct the candidate ignoring the first observations. The candidate Σcandc is therefore sampled

from an inverted Wishart distribution with scale parameter∑Tc

t=2 εctε′ct and Tc − 1 degrees of

freedom, where εct = ∆ log Sct−µc−∑K

k=1 Ak∆log Xckt−Πc(log Sc,t−1−∑K

k=1 Bck∆log Xck,t−1).

The acceptance-rejection probability equalsφ(ε1c; 0, Vc)|Σc=Σcand

c

φ(ε1c; 0, Vc)|Σc=Σoldc

, (20)

where Σoldc denotes the previous draw of Σc.

Sampling of ΛnΛnΛn and Θn, n = 1, 2Θn, n = 1, 2Θn, n = 1, 2

The full conditional distributions of Λn and Θn are all multivariate normal. Below we present

the derivation for Λ1. Next we summarize the parameters of the multivariate normal for Λ2 and

Θn, n = 1, 2 in a table. To sample Λ1, we note that we can write (11) as

α′ic = h′icΛ1 + ν1ic′, (21)

and hence it is a multivariate regression model with regression matrix Λ1. Hence, the full

conditional posterior distribution of Λ1 is a matrix normal distribution with mean(

C∑

c=1

Ic∑

i=1

hich′ic

)−1 (C∑

c=1

Ic∑

i=1

hicαic

), (22)

and covariance matrix Σ1

ν ⊗(

C∑

c=1

Ic∑

i=1

hich′ic

)−1 . (23)

The corresponding means and covariance matrices for the other full conditional distributions

are given belowParameter Mean Covariance matrix

Λ2

(∑Cc=1

∑Ici=1 hich

′ic

)−1 (∑Cc=1

∑Ici=1 hicβic

) (Σ2

ν ⊗(∑C

c=1

∑Ici=1 hich

′ic

)−1)

Θ1

(∑Cc=1

∑Ic(Ic−1)l=1 ZlcZ

′lc

)−1 (∑Cc=1

∑Ic(Ic−1)l=1 Zlcαlc

) (Σ1

η ⊗(∑C

c=1

∑Ic(Ic−1)l=1 ZlcZ

′lc

)−1)

Θ2

(∑Cc=1

∑Ic(Ic−1)l=1 ZlcZ

′lc

)−1 (∑Cc=1

∑Ic(Ic−1)l=1 Zlcβlc

) (Σ2

η ⊗(∑C

c=1

∑Ic(Ic−1)l=1 ZlcZ

′lc

)−1)

29

Sampling of ΣnνΣnνΣnν and Σn

ηΣnηΣnη , n = 1, 2n = 1, 2n = 1, 2

To sample these covariance matrices we note that (11) to (14) are a multivariate regression

models. Hence the full conditional posterior distribution of are inverted Wishart distribution

with scale parameter and degrees of freedom as defined in the overview below.Covariance Scale parameter Degrees of freedom

Σ1ν κ1IK +

∑Cc=1

∑Ici=1(αic − Λ′1zic)(αic − Λ′1hic)′ κ2 +

∑Cc=1 Ic

Σ2ν κ1IK +

∑Cc=1

∑Ici=1(βic − Λ′2hic)(βic − Λ′2gic)′ κ2 +

∑Cc=1 Ic

Σ1η κ1IK +

∑Cc=1

∑Ic(Ic−1)l=1 (αlc −Θ′

1Zlc)(αlc −Θ′1Zlc)′ κ2 +

∑Cc=1 Ic(Ic − 1)

Σ2η κ1IK +

∑Cc=1

∑Ic(Ic−1)l=1 (βlc −Θ′

2Zlc)(βlc −Θ′2Zlc)′ κ2 +

∑Cc=1 Ic(Ic − 1)

The κ terms results from the inverted Wishart priors which are used to improve convergence of

our Gibbs sampler (Hobert and Casella, 1996).

Sampling of αcαcαc and βcβcβc

First we need to define some additional notation. We split up Xckt = (Xck1t, . . . , XckIct)′ for

k = 1, . . . ,K into two parts Xowncit = [Xckit]Kk=1 and Xcross

cit = [[Xckjt]Kk=1]Icj=16=i to disentangle the

own effects from the cross effects. Note that Xowncit and Xcross

cit are both row vectors. Now further

define Xownct = diag(Xown

c1t , . . . , XowncIct ) and Xcross

ct = diag(Xcrossc1t , . . . , Xcross

cIct ).

To sample αc and βc jointly we rewrite the second equation of (30) as

log Sc1 − log Xcrossc1 βc + Π−1

c µc =(0 log Xown

c1

)(αc

βc

)+ εc1

∆log Sct − µc −∆log Xcrossct αc −Πc(log Sc,t−1 − log Xcross

c,t−1βc) =(∆log Xown

ct −Πc log Xownc,t−1

)(αc

βc

)+ εct,

(24)

which we write in matrix notation as

Yct = Wct

(αc

βc

)+ εct, (25)

with the obvious definitions for Yct as the left-hand side of the equation and Wct the matrix

appearing on the right-hand side. The variance of εc1 is Vc and the variance of εct, t > 1 equals

Σc. Note that we use different definitions for Wct and Yct for t = 1 versus t > 1. Next we write

the Ic equations of (11) and (12) as

−U1c = −IKIcαc + ν1

c

−U2c = −IKIcβc + ν2

c ,(26)

30

where Unc is a (KIc)-dimensional vector containing the terms Λ′nhic, i = 1, . . . , Ic, for n = 1, 2

and where IKIc is a (KIc)-dimensional identity matrix. The error term νnc is normal distributed

with mean 0 and covariance matrix (IIc ⊗ Σnν ), n = 1, 2. To sample αc and βc, we combine and

standardize (25) and (26)

Vc− 1

2 Yc1 = Vc− 1

2 Wc1

(αc

βc

)+ Vc

− 12 εc1

Σc− 1

2 Yct = Σc− 1

2 Wct

(αc

βc

)+ Σc

− 12 εct

−(IIc ⊗ Σ1

ν− 1

2 )U1c

−(IIc ⊗ Σ2ν− 1

2 )U2c

=

−(IIc ⊗ Σ1

ν− 1

2 ) 0

0 −(IIc ⊗ Σ2ν− 1

2 )

(αc

βc

)+

(IIc ⊗ Σ1

ν− 1

2 )ν1c

(IIc ⊗ Σ2ν− 1

2 )ν2c

,

(27)

where A−12 denotes the inverse of the Choleski decomposition of the matrix A, that is A−

12 =

(A12 )−1 with A

12 A

12′= A. Hence, the full conditional posterior distribution of (α′c, β′c)′ is normal

with mean

(((IIc ⊗ Σ1

ν−1) 0

0 (IIc ⊗ Σ2ν−1)

)+ W ′

c1V−1c Wc1 +

Tc∑

t=2

(W ′ctΣ

−1c Wct)

)−1

(((IIc ⊗ Σ1

ν−1)U1

c

(IIc ⊗ Σ2ν−1)U2

c

)+ (W ′

c1V−1c Yc1) +

Tc∑

t=2

(W ′ctΣ

−1c Yct)

), (28)

and covariance matrix((

(IIc ⊗ Σ1ν−1) 0

0 (IIc ⊗ Σ2ν−1)

)+ W ′

c1V−1c Wc1 +

Tc∑

t=2

(W ′ctΣ

−1c Wct)

)−1

(29)

Sampling of µcµcµc, αcαcαc and βcβcβc

Equation (10) and (1) can now be written as

log Sc1 − log Xownc1 βc =

(−Π−1

c 0 log Xcrossc1

)

µc

αc

βc

+ εc1

∆log Sct −∆log Xownct αc

−Πc(log Sc,t−1 − log Xownc,t−1βc) =

(IIc ∆log Xcross

ct −Πc log Xcrossc,t−1

)

µc

αc

βc

+ εct,

(30)

31

where αc and βc capture the cross-effects in the matrices Ack and Bck for k = 1, . . . , K. This

system can be written in a multivariate regression model

Yct = Wctγ + εct, (31)

where Yct contains the left-hand side of (30), Wct contains (−Π−1c

...0... log Xcross

c1 ) for the first

observation and (Ic...∆ log Xcross

ct

...−Πc log Xcrossc,t−1) for the remaining observations, and where γ =

(µ′c, α′c, β′c)′. The error term is normal distributed with mean 0 and covariance matrix Σc (and

Vc for the first observation).

In the same manner as before we collect and standardize the information from the second

layer of the hierarchical model.

−U1c = −IKIc(Ic−1)αc + η1

c

−U2c = −IKIc(Ic−1)βc + η2

c ,(32)

where Unc stacks the Ic(Ic − 1) vectors Θ′

nZlc, n = 1, 2.

Vc− 1

2 Yc1 = Vc− 1

2 Wc1γ + Vc− 1

2 εc1

Σc− 1

2 Yct = Σc− 1

2 Wctγ + Σc− 1

2 εct−(IIc(Ic−1) ⊗ Σ1

η− 1

2 )U1c

−(IIc(Ic−1) ⊗ Σ2η− 1

2 )U2c

=

0 −(IIc(Ic−1) ⊗ Σ1

η− 1

2 ) 0

0 0 −(IIc(Ic−1) ⊗ Σ2η− 1

2 )

γ+

+

(IIc(Ic−1) ⊗ Σ1

η− 1

2 )η1c

(IIc(Ic−1) ⊗ Σ2η− 1

2 )η2c

.

(33)

Hence, the full conditional distribution of γ is normal with mean

0 0 0

0 (IIc(Ic−1) ⊗ Σ1η−1) 0

0 0 (IIc(Ic−1) ⊗ Σ2η−1)

+ W ′

c1V−1c Wc1 +

Tc∑

t=2

(W ′ctΣ

−1c Wct)

−1

0

(IIc(Ic−1) ⊗ Σ1η−1)U1

c

(IIc(Ic−1) ⊗ Σ2η−1)U2

c

+ (W ′

c1V−1c Yc1) +

Tc∑

t=2

(W ′ctΣ

−1c Yct)

, (34)

and covariance matrix

0 0 0

0 (IIc(Ic−1) ⊗ Σ1η−1) 0

0 0 (IIc(Ic−1) ⊗ Σ2η−1)

+ W ′

c1V−1c Wc1 +

Tc∑

t=2

(W ′ctΣ

−1c Wct)

−1

. (35)

32

B The definition of explanatory variables (zij,czij,czij,c)

In this appendix we describe the characteristics that we use to explain the cross-price

effects. If necessary we give a formal mathematical definition. We organize the charac-

teristics based on the level at which they are defined (category level, brand level or both)

and the concept they measure (eg. competitive intensity). For the sake of comparion we

closely follow Fok et al. (2006) for the choice and definitions of the variables.

In this appendix we use the following notation:

Sict Sales volume of brand i in category c at time t

Mict = Sict/∑Ic

i=1 Sict Market share of brand i at time t

M ic = 1Tc

∑Tc

t=1 Mict (Time) average market share

Pict (Actual) price of brand i in category c at time t

RPict Regular price of brand i in category c at time t

RP c = 1IcTc

∑Tc

t=1

∑Ic

i=1 RPict Average regular price in category c

PIict = Pit/RPit (Promotional) Price index

The explanatory variables are defined as follows

Category-Specific Characteristics

Average budget share 1Tc

∑Tc

t=1

∑Ic

i=1 SictPict

Utilitarian low (0), middle (0.5), high (1) (defined in Fok et al.

(2006))

Perishability low (0), middle (0.5), high (1) (defined in Fok et al.

(2006))

Market concentration∑Ic

i=1 M ic log M ic, see Raju (1992)

Price dispersion∑Tc

t=1 (maxi (RPict)−mini (RPict)) /(Tc ·RP c

)

Category- and Brand-Specific Characteristics

Price promotion frequency percentage of observations where price index is below

0.95.

Depth of price promotions∑Tc

t=1 log(PIit)

FREQic, where FREQic denotes the price promo-

tion frequency of brand i in category c.

Feature/Display frequency

(brand level)

average of the percentage of SKUs promoted by the

brand over time.

Feature/Display frequency

(category level)

∑Tct=1 1−∏Ic

i=1(1−xit)

Tc, where xict denotes the percentage of

SKUs promoted by brand i in category c at time t.

33

Brand-Specific Characteristics

Similarity of price Absolute distance between attacker and victim brand’s

average regular price relative to the average category

regular price

Similarity of size absolute distance between attacker and victim average

market share

Asymmetric price effect indicator variable that equals one if the attacker brand

is more expensive than the victim brand, zero otherwise.

Asymmetric size effect indicator variable that equals one if the attacker brand

has a higher average market share than the victim

brand, zero otherwise.

34

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Publications in the Report Series Research∗ in Management ERIM Research Program: “Marketing” 2008 Experts' Stated Behavior Youssef Boulaksil and Philip Hans Franses ERS-2008-001-MKT http://hdl.handle.net/1765/10900 The Value of Analogical Reasoning for the Design of Creative Sales Promotion Campaigns: A Case-Based Reasoning Approach Niek A.P. Althuizen and Berend Wierenga ERS-2008-006-MKT http://hdl.handle.net/1765/11289 Shopping Context and Consumers' Mental Representation of Complex Shopping Trip Decision Problems Benedict G.C. Dellaert, Theo A. Arentze and Harry J.P. Timmermans ERS-2008-016-MKT http://hdl.handle.net/1765/11812 Modeling the Effectiveness of Hourly Direct-Response Radio Commercials Meltem Kiygi Calli, Marcel Weverbergh and Philip Hans Franses ERS-2008-019-MKT http://hdl.handle.net/1765/12242 Choosing Attribute Weights for Item Dissimilarity using Clikstream Data with an Application to a Product Catalog Map Martijn Kagie, Michiel van Wezel and Patrick J.F. Groenen ERS-2008-024-MKT http://hdl.handle.net/1765/12243 The Effect of Superstar Software on Hardware Sales in System Markets Jeroen L.G. Binken and Stefan Stremersch ERS-2008-025-MKT http://hdl.handle.net/1765/12339 Sales Growth of New Pharmaceuticals Across the Globe: The Role of Regulatory Regimes Stefan Stremersch and Aurélie Lemmens ERS-2008-026-MKT http://hdl.handle.net/1765/12340 When Intelligence is (Dys)Functional for Achieving Sales Performance Willem J. Verbeke, Frank D. Belschak, Arnold B. Bakker, and Bart Dietz ERS-2008-034-MKT http://hdl.handle.net/1765/12633 Path Dependencies and the Long-term Effects of Routinized Marketing Decisions Paul Farris, Willem J. Verbeke, Peter Dickson and Erjen van Nierop ERS-2008-035-MKT http://hdl.handle.net/1765/12634 Does Irritation Induced by Charitable Direct Mailings Reduce Donations? Merel van Diepen, Bas Donkers and Philip Hans Franses ERS-2008-036-MKT http://hdl.handle.net/1765/12704

Brain Mechanisms of Persuasion: How “Expert Power” Modulates Memory and Attitudes Vasily Klucharev, Ale Smidts and Guillén Fernández ERS-2008-038-MKT http://hdl.handle.net/1765/12784 Moderating Factors of Immediate, Dynamic, and Long-run Cross-Price Effects Csilla Horváth and Dennis Fok ERS-2008-042-MKT http://hdl.handle.net/1765/12901 Why, How and When Do Prices Land? Evidence from the Videogame Industry Carlos Hernandez-Mireles, Dennis Fok and Philip Hans Franses ERS-2008-044-MKT http://hdl.handle.net/1765/12900 ∗ A complete overview of the ERIM Report Series Research in Management:

https://ep.eur.nl/handle/1765/1 ERIM Research Programs: LIS Business Processes, Logistics and Information Systems ORG Organizing for Performance MKT Marketing F&A Finance and Accounting STR Strategy and Entrepreneurship